arXiv:2509.06037v3

Polynomial entropy of induced homeomorphisms on $C_k(\mathbb{S}^1)$ and $C_k([0,1])$

Maša Đorić, Jelena Katić

math.DS37B4054F1637A35

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Paper metadata

Original submission: September 7, 2025
Latest revision: August 3, 2026 · version 3
Claimed MSC: 37B40, 54F16, 37A35
arXiv categories: math.DS
Paper license: arXiv non-exclusive distribution license ↗

Abstract

We compute the polynomial entropy of $C_k(f)$ where $f$ is any circle or interval homeomorphism.

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Audit summary

Audited against arXiv v3

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements2 reported findingsContains wrong statements

The circle alternative in Theorem 2 is false for orientation-reversing involutions: a reflection is not conjugate to a rotation, but its induced map has polynomial entropy zero rather than $2k$ . The rotation-conjugate zero-entropy branch is correct.

Theorem 2 · circle caseIncorrect

A circle reflection contradicts the stated dichotomy

Pages 1–2 and 8 · Theorem 2 and its proof · arXiv:2509.06037v3

The theorem claims that every circle homeomorphism not conjugate to a rotation satisfies $h_{\mathrm{pol}}(C_k(f))=2k$ . Let $f$ be a reflection of $S^1$ . Orientation type is invariant under conjugacy, so this orientation-reversing map is not conjugate to any rotation. But $f^2=\operatorname{Id}$ , hence $C_k(f)^2=C_k(f^2)=\operatorname{Id}.$ The separated-set cardinalities for $C_k(f)$ are therefore bounded independently of the orbit length once the two iterates have been included, and $h_{\mathrm{pol}}(C_k(f))=0$ . This is a counterexample for every $k\geq1$ . A substantive repair must change the circle statement, for example by imposing orientation preservation; no local proof correction can make the printed dichotomy true.

Reviewed manuscript, version 3 ↗

Theorem 2 · rotation branchCorrect

Conjugacy to a rotation gives zero polynomial entropy

Pages 3 and 8 · Lemma 3 and proof of Theorem 2 · arXiv:2509.06037v3

A rotation of $S^1$ is an isometry, and its induced action on $C_k(S^1)$ preserves the Hausdorff metric. Polynomial entropy is invariant under topological conjugacy, while an isometry has bounded separated-set growth. Thus the stated zero-entropy conclusion in this branch follows. The printed proof of Lemma 3 has a repairable inference defect recorded separately in the proof audit, but the lemma and this branch are correct by the verified inverse-isometry argument given there.

02Proofs4 reported findingsContains incorrect or incomplete proofs

The proof discards the orientation-reversing case by passing to $f^2$ , although this loses the distinction needed by the theorem and admits the reflection counterexample. Independently, Proposition 7 relies on the false assertion that taking boundaries is $1$ -Lipschitz in Hausdorff distance. Lemma 3 contains a local false inference but has a fully verified repair.

Proof of Theorem 2 · orientation reductionIncorrect as written

Passing to the square does not preserve the theorem's case distinction

Page 8 · first paragraph of the proof of Theorem 2 · arXiv:2509.06037v3

The proof says that one may assume $f$ orientation-preserving because $f^2$ is orientation-preserving and $h_{\mathrm{pol}}(C_k(f^2))=h_{\mathrm{pol}}(C_k(f))$ . The entropy identity is valid, but it does not imply that the printed hypothesis branch for $f^2$ is the same as that for $f$ . For a reflection, $f^2=\operatorname{Id}$ is conjugate to a rotation while $f$ is not, so the argument gives entropy zero and contradicts the theorem's claimed $2k$ . Repair classification: No repair supplied. The theorem itself must be restricted or its zero-entropy alternative enlarged.

Proposition 7Incorrect as written

The boundary map is not $1$ -Lipschitz

Pages 7–8 · proof of Proposition 7, after Equation (2) · arXiv:2509.06037v3

The lower bound is deduced from the assertion that $\pi(K)=\partial K$ is $1$ -Lipschitz from $C_k(S^1)\setminus\{S^1\}$ to $F_{2k}(S^1)$ . This assertion is false for $k\geq2$ . Inside an embedded arc identified with $[0,1]$ , take $K=[0,1],\qquad K_\varepsilon=[0,1/2]\cup[1/2+\varepsilon,1].$ Then $d_H(K,K_\varepsilon)=\varepsilon/2$ , whereas $\partial K_\varepsilon$ contains points near $1/2$ whose distance from $\partial K=\{0,1\}$ stays close to $1/2$ . Consequently the displayed inequality comparing Hausdorff distances of the lifts with those of their boundaries can fail. Proposition 8 repeats the same argument, so the printed lower-bound proofs for the nonzero-entropy branches of Theorem 2 are not established by this route. Repair classification: No repair supplied; a different separated-set construction or another verified lower-bound argument is required.

Reviewed manuscript, version 3 ↗

Lemma 3Incorrect as written

The reverse Hausdorff inequality does not follow from an arbitrary distant pair

Page 3 · final two sentences of the proof of Lemma 3 · arXiv:2509.06037v3

From $s<d_H(A,B)$ , the proof selects $a\in A$ and $b\in B$ with $d(a,b)>s$ and concludes that $f(A)\not\subset U(f(B),s)$ . One distant pair does not establish that $f(a)$ is farther than $s$ from every point of $f(B)$ , so this inference is false. Repair classification: Verified repair. The preceding argument already proves $d_H(f(A),f(B))\leq d_H(A,B).$ Apply the same inequality to the inverse isometry $f^{-1}$ and the sets $f(A),f(B)$ to obtain the reverse inequality. This proves the lemma and repairs every later use without changing any statement.

Proposition 8Minor formal correction

The regularity hypothesis is mismatched with Theorem 2

Page 8 · statement of Proposition 8 and final sentence of the proof of Theorem 2 · arXiv:2509.06037v3

Proposition 8 is printed for an interval diffeomorphism, but Theorem 2 is stated for every interval homeomorphism and invokes Proposition 8 directly. The argument in Proposition 8 uses only that an interval homeomorphism is monotone and that its square is increasing; it uses no differentiability. Replace `diffeomorphism' by `homeomorphism' in Proposition 8. This uniquely determined local change fixes the hypothesis mismatch, although it does not repair the separate false boundary-map step inherited from Proposition 7.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2509.06037v3
Authors listed: Maša Đorić, Jelena Katić
Audit date: August 18, 2026

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For each proof or proof segment, use the narrowest supported outcome: - Correct and complete: every substantive step is verifiable and no case or crucial nontrivial step is missing. - Incorrect as written: a displayed calculation, inference, lemma, or cited application is formally false or inapplicable. - Incomplete as written: a necessary substantive step or case is missing, even though no explicit contradiction has been proved. - Not able to verify: the available paper and sources are insufficient to decide whether the step is valid. Do not flag omitted routine details, compressed arguments, or “analogously” steps when the omitted reasoning is recoverable and correct. 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If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. 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P(delta)

is proved for every sufficiently small

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P

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P(delta)

holds for every

delta>0

requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

and subsequently applied at

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Polynomial entropy of induced homeomorphisms on Ck(S1)C_k(\mathbb{S}^1)Ck​(S1) and Ck([0,1])C_k([0,1])Ck​([0,1])

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

A circle reflection contradicts the stated dichotomy

Conjugacy to a rotation gives zero polynomial entropy

Passing to the square does not preserve the theorem's case distinction

The boundary map is not 111-Lipschitz

The reverse Hausdorff inequality does not follow from an arbitrary distant pair

The regularity hypothesis is mismatched with Theorem 2

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Polynomial entropy of induced homeomorphisms on $C_k(\mathbb{S}^1)$ and $C_k([0,1])$

The boundary map is not $1$ -Lipschitz